I hope the second answer I have just posted was less discouraging than my first. But you were less than positive yourself about the transparency of what you were driving at. Your title 'Dog, hog, off etc.' made that perfectly obvious. Do the others have different vowels in them, I wonder. That would sort of prove your point, wouldn't it?
I was born ancient, and surrounded by ancients, so I had the royal 'awf' until I ditched it.* Now I have the same vowel in all three, as no doubt you will tell me you do, but everywhere else as well.
What may indeed not be obvious from your question is that it also involves the q of whether these spellings with 'au' (or of course 'aw', 'al' ,'all'. augh. ough. etc. ) are thrown in with all these other Os.
*BTW, the Queen herself is said to be demoticizing down. Have you observed whether she has kept all her aw's? Must listen more carefully. Her grandfather "lahnched" ships, I learnt from an old newsreel. Even her mother merely "launched" a ship, "and all who sail inner".
We are all on the slippery slope, which is why I at first did not recommend conservation!